What about such map from a non compact set to compact set in nice topologies?
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AlexMay 31 '11 at 12:14

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what exactly is the reason for non existence? In the (0,1) case some sort of local compactness is the reason.Can the proof generalized to non existence of a map from open ball in R^n to a closed ball.
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AlexMay 31 '11 at 12:27

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@Alex: By the way, since you're new here: you can accept an answer by clicking on the grey checkmark sign to the right of it (I'm not saying you should accept mine even though I wouldn't complain).
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t.b.May 31 '11 at 12:53

5 Answers
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No. If $f:(0,1) \to [0,1]$ were continuous and bijective, there would be a unique point $x \in (0,1)$ such that $f(x) = 1$. However, since $f$ is continuous, the intervals $[x - \varepsilon, x]$ and $[x, x + \varepsilon]$ would be mapped to intervals $[a,1]$ and $[b,1]$, say. By bijectivity we'd have $a, b \lt 1$. Thus every value strictly between $\max{\{a,b\}}$ and $1$ would be assumed at least twice, contradicting bijectivity.

Let $f:(0,1) \rightarrow [0,1]$ be continuous and surjective. (Actually, we just need to suppose that $0$ and $1$ are in the image of $f$.) Let $a,b \in (0,1)$ such that $f(a)=0$ and $f(b)=1$. Let $I=[a,b]$ if $a<b$ or $I=[b,a]$ if $b<a$. Then, by the intermediate value theorem, $f(I)$ is an interval that contains $0$ and $1$ and so $f(I)$ contains $[0,1]$, which implies $f(I)=[0,1]$. But then $f$ cannot be injective because there are lots of points in $(0,1)\setminus I$.

Suppose that $f:(0,1) \rightarrow [0,1]$ is 1-1 and continuous. By the intermediate value theorem, the image of any interval under $f$ is an interval. Since $f$ is 1-1, it is either (strictly) monotone increasing or decreasing. Hence, $f(0,1)$ is an interval. Without loss of generality, assume $f$ is increasing; were it not this analysis would apply to $1 - f$.

Suppose now that $f$ is onto; then we must have some $t\in(0,1)$ with $f(t) = 1$. Because $f$ is strictly monotone increasing, we would have to have $f(s) > 1$, for $t \le s < 1$. This violates the premise that $f(0,1)
\subseteq [0,1]$. Hence, $f$ cannot be onto.

Since Theo gave an answer I am going to be nitpicking and add one remark. When speaking about continuity (especially when tagging under [topology]) it is best to mention the topology you are working with. In this case, you mean in the standard topology.

Otherwise, consider the discrete topology, i.e. every set is open:

Let $f\colon [0,1]\to (0,1)$ be any bijection, it is continuous since all sets are open, the preimage of an open set is an open set, thus $f$ is continuous.

Oy, do you really want me not to downvote this? :) Of course, you're right, but come on! :), +1 for the effort
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t.b.May 31 '11 at 11:49

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On the contrary: when mentioning subsets of the reals, assume the standard topology unless otherwise stated.
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GEdgarMay 31 '11 at 13:24

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@GEdgar: When taking a course in real analysis? Sure. When taking a course in point-set topology? Not if you want to be accurate.
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Asaf KaragilaMay 31 '11 at 14:00

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@Asaf: When answering a question on this board? YES. Or, if you want to be super-accurate, add "assuming the usual topology" to your answer.
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GEdgarJun 1 '11 at 13:25

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@AsafKaragila: it would be complete to give an example of one such bijection (e.g. $0\mapsto\frac12$, $\frac1n\mapsto\frac1{n+2}$ for $n=1,2,3,\dots$, and $x\mapsto x$, otherwise).
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robjohn♦Nov 4 '12 at 15:28

There does not exist a continuous bijection from (0,1) to [0,1]. Indeed, let $f$ be such a function. Let consider a sequence $x_n=1-1/n$. Then from the sequence $(f(x_n))$ we can choose a subsequence $(f(x_{n_k}))$ which is convergent. Let denote this limit by $y$. Obviously, $y \in [0,1]$. Since $f^{-1}$ also is continuous, we get $f^{-1}(y)=\lim_{k \to +\infty}f^{-1}(f(x_{n_k}))=\lim_{k \to \infty}x_{n_k}=1$. But $1 \notin (0,1)$.

Remark(Why $f^{-1}$ must be continuous under our assumption?) By our assumption $f:(0,1)\to [0,1]$ is continuous bijection. Then $f:(0,1)\to [0,1]$ must be injective and continuous which following invariance of domain (see, http://en.wikipedia.org/wiki/Invariance_of_domain is homeomorphism. Hence $f^{-1}: [0,1]\to (0,1)$ is continuous.

In order to prove that $f^{-1}$ is continuous under our assumption, I use only one argument of the invariance of domain at en.wikipedia.org/wiki/Invariance_of_domain asserted that if $U$ is open subset of $R^n$ and $f:U\to R^n$ is injective and continuous that $f(U)$ is open and $f:U \to f(U)$ is homeomorphism. My previous comment assumes this situation.
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Gogi PantsulaiaAug 15 '14 at 9:26

That argument should be made in the answer (although invariance of domain is serious overkill for the one-dimensional setting).
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Daniel Fischer♦Aug 15 '14 at 12:37